Weak limits for exploratory plots in the analysis of extremes

TL;DR

This paper analyzes the asymptotic behavior of quantile-quantile and mean excess plots in extreme value analysis, providing distributional limits and confidence bounds to test for heavy-tailed distributions.

Contribution

It derives asymptotic distributional limits for these plots under regularly varying tails and introduces confidence bounds for statistical testing of heavy tails.

Findings

Asymptotic distributional limits are established for the plots.

Confidence bounds enable statistical testing for heavy tails.

Results apply to distributions with regularly varying right-tails.

Abstract

Exploratory data analysis is often used to test the goodness-of-fit of sample observations to specific target distributions. A few such graphical tools have been extensively used to detect subexponential or heavy-tailed behavior in observed data. In this paper we discuss asymptotic limit behavior of two such plotting tools: the quantile-quantile plot and the mean excess plot. The weak consistency of these plots to fixed limit sets in an appropriate topology of $\mathbb{R}^2$ has been shown in Das and Resnick (Stoch. Models 24 (2008) 103-132) and Ghosh and Resnick (Stochastic Process. Appl. 120 (2010) 1492-1517). In this paper we find asymptotic distributional limits for these plots when the underlying distributions have regularly varying right-tails. As an application we construct confidence bounds around the plots which enable us to statistically test whether the underlying distribution is heavy-tailed or not.